快速排序法02:普通快速排序法的优化
随机化选择标定点
为了解决对有序数组排序,快速排序法复杂度变成O(n^2)的问题,在partition()方法中随机化选择一个元素作为标定点
import java.util.Arrays;
import java.util.Random;
public class Algorithm {
public static void main(String[] args) {
Integer[] arr = {3, 2, 5, 1, 0};
QuickSort.sortOptimized(arr);
System.out.println(Arrays.toString(arr));
}
}
class QuickSort {
private QuickSort(){}
public static<E extends Comparable<E>> void sortOptimized(E[] arr){
/**
* 将重复使用的temp和random变量作为参数传递,进行内存优化
*/
Random random = new Random();
E temp = null;
sortOptimized(arr, 0, arr.length - 1, random, temp);
}
public static<E extends Comparable<E>> void sortOptimized(E[] arr, int left, int right, Random random, E temp){
if (left >= right){
return;
}
int p = partition(arr, left, right, random, temp);
sortOptimized(arr, left, p - 1, random, temp);
sortOptimized(arr, p + 1, right, random, temp);
}
public static <E extends Comparable<E>> int partition(E[] arr, int left, int right, Random random, E temp){
/**
* 生成[left, right]之间的随机索引,然后和left互换位置作为第一个元素
* random.nextInt()方法只能生成[0, bound)的整数,而分区间是以left起始的,长度为right - left + 1,所以在left的基础上,再生成[0, right - left + 1)范围内的整数
*/
int p = random.nextInt(right - left + 1) + left;
temp = arr[p];
arr[p] = arr[left];
arr[left] = temp;
int j = left;
for (int i = left + 1; i <= right; i++) {
if (arr[i].compareTo(arr[left]) < 0){
j++;
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
temp = arr[j];
arr[j] = arr[left];
arr[left] = temp;
return j;
}
}
练习:对于总是将中间的元素作为标定点的快速排序,编写一个算例,生成一个中间元素总是最小的数组,即让这个快速排序法“失效”
import java.util.Arrays;
import java.util.Random;
public class Algorithm {
public static void main(String[] args) {
Integer[] testScale = {10000, 100000};
for (Integer n : testScale) {
Integer[] randomArr = ArrayGenerator.generatorRandomArray(n, n);
Integer[] sortedArr = ArrayGenerator.generatorSortedArray(n);
Integer[] specialArr = ArrayGenerator.generateSpecialArray(n);
Integer[] arr1 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr3 = Arrays.copyOf(randomArr, randomArr.length);
Integer[] arr2 = Arrays.copyOf(sortedArr, sortedArr.length);
Integer[] arr4 = Arrays.copyOf(sortedArr, sortedArr.length);
Integer[] arr5 = Arrays.copyOf(specialArr, specialArr.length);
Integer[] arr6 = Arrays.copyOf(specialArr, specialArr.length);
System.out.println("测试随机数组排序性能");
System.out.println();
Verify.testTime("QuickSortMid", arr1);
Verify.testTime("QuickSortOptimized", arr3);
System.out.println();
System.out.println("测试有序数组排序性能");
System.out.println();
Verify.testTime("QuickSortMid", arr2);
Verify.testTime("QuickSortOptimized", arr4);
System.out.println();
System.out.println("测试特殊数组排序性能");
System.out.println();
Verify.testTime("QuickSortMid", arr5);
Verify.testTime("QuickSortOptimized", arr6);
System.out.println();
}
}
}
class QuickSort {
private QuickSort(){}
public static<E extends Comparable<E>> void sortMid(E[] arr){
E temp = null;
sortMid(arr, 0, arr.length - 1, temp);
}
public static<E extends Comparable<E>> void sortMid(E[] arr, int left, int right, E temp){
if (left >= right){
return;
}
int p = partition(arr, left, right, temp);
sortMid(arr, left, p - 1, temp);
sortMid(arr, p + 1, right, temp);
}
public static <E extends Comparable<E>> int partition(E[] arr, int left, int right, E temp){
int mid = left + (right - left) / 2;
temp = arr[mid];
arr[mid] = arr[left];
arr[left] = temp;
int j = left;
for (int i = left + 1; i <= right; i++) {
if (arr[i].compareTo(arr[left]) < 0){
j++;
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
temp = arr[j];
arr[j] = arr[left];
arr[left] = temp;
return j;
}
public static<E extends Comparable<E>> void sortOptimized(E[] arr){
Random random = new Random();
E temp = null;
sortOptimized(arr, 0, arr.length - 1, random, temp);
}
public static<E extends Comparable<E>> void sortOptimized(E[] arr, int left, int right, Random random, E temp){
if (left >= right){
return;
}
int p = partitionOptimized(arr, left, right, random, temp);
sortOptimized(arr, left, p - 1, random, temp);
sortOptimized(arr, p + 1, right, random, temp);
}
public static <E extends Comparable<E>> int partitionOptimized(E[] arr, int left, int right, Random random, E temp){
int p = random.nextInt(right - left + 1) + left;
temp = arr[p];
arr[p] = arr[left];
arr[left] = temp;
int j = left;
for (int i = left + 1; i <= right; i++) {
if (arr[i].compareTo(arr[left]) < 0){
j++;
temp = arr[i];
arr[i] = arr[j];
arr[j] = temp;
}
}
temp = arr[j];
arr[j] = arr[left];
arr[left] = temp;
return j;
}
}
class ArrayGenerator {
private ArrayGenerator() {}
public static Integer[] generatorRandomArray(Integer n, Integer maxBound) {
Integer[] arr = new Integer[n];
Random random = new Random();
for (int i = 0; i < n; i++) {
arr[i] = random.nextInt(maxBound);
}
return arr;
}
public static Integer[] generatorSortedArray(Integer n) {
Integer[] arr = new Integer[n];
for (int i = 0; i < n; i++) {
arr[i] = i;
}
return arr;
}
public static Integer[] generateSpecialArray(int n) {
Integer[] arr = new Integer[n];
generateSpecialArray(arr, 0, n - 1, 0);
return arr;
}
public static void generateSpecialArray(Integer[] arr, int left, int right, int value) {
if (left >= right) {
return;
}
int mid = left + (right - left) / 2;
/**
* 生成中间值总是最小的数组,就是反向模拟partition()这个过程
* partition()过程是将中间值和第一个元素互换,我们的目的是让这个中间值最小,这样换了以后,下一个左区间没有元素,每次只能将第一个最小的元素排好序
* 反向模拟,就是先让数组的中间元素最小,然后和第一个元素互换,再对剩下的[left + 1, right]区间进行同样的递归
*/
arr[mid] = value;
int temp;
temp = arr[mid];
arr[mid] = arr[left];
arr[left] = temp;
generateSpecialArray(arr, left + 1, right, value + 1);
temp = arr[mid];
arr[mid] = arr[left];
arr[left] = temp;
/**
* 因为mid是左区间最后一个元素,因此左区间的元素都会被赋值,可是右区间最后一个元素可能不会被递归到,因为mid的取值是向下取整
* 为了防止最后一个元素没有被赋值,所以要额外判断一下,如果没有遍历到,可以赋值一个大于value的值
*/
if (arr[right] == null){
arr[right] = value + 2;
}
}
}
class Verify {
private Verify (){}
public static<E extends Comparable<E>> boolean isSorted(E[] arr){
for (int i = 0; i < arr.length - 1; i++) {
if (arr[i].compareTo(arr[i + 1]) > 0) {
return false;
}
}
return true;
}
public static<E extends Comparable<E>> void testTime(String AlgorithmName, E[] arr) {
long startTime = System.nanoTime();
if (AlgorithmName.equals("QuickSortMid")) {
QuickSort.sortMid(arr);
}
if (AlgorithmName.equals("QuickSortOptimized")) {
QuickSort.sortOptimized(arr);
}
long endTime = System.nanoTime();
if (!Verify.isSorted(arr)){
throw new RuntimeException(AlgorithmName + "算法排序失败!");
}
System.out.println(String.format("%s算法,测试用例为%d,执行时间:%f秒", AlgorithmName, arr.length, (endTime - startTime) / 1000000000.0));
}
}
普通快速排序法还有问题
在目前的partition()方法,数组的元素越无序,快速排序法的性能越好。虽然随机化选择标定点对有序的数组也有了性能的提升
但还有一个问题:如果数组的所有元素都一样,那就等价于是完全有序数组,而且随机化标定点的策略也会失效,此时快速排序法的复杂度仍会降到O(n^2)